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	<h1>REML estimation; <br>
	Assessment of BCB bowhead whales </h1>
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                     Code: <a href="bcb.tpl">bcb.tpl</a><br>
                     Data: 		<a href="bcb.dat">bcb.dat</a><br>
                     Initial values: <a href="bcb.pin">bcb.pin</a><br>
<!--
                     All required files (DOS): <a href="bcb.zip">bcb.zip</a><br>
					 All required files (linux): <a href="bcb.tar.gz">bcb.tar.gz</a><br>					 
-->
                     Expected Results: <a href="bcb-expected-results.par">bcb.par</a><br>
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<h3><strong>Model description</strong></h3>

Restricted maximum likelihood estimation (REML) is often used to estimate variance parameters in the linear
mixed-model:
<br>
<div align="center">
	<em>y</em> = <em>X*b</em> + <em>Z*u</em> + <em>e</em>
</div>
where <em>X</em> and <em>Z</em> are covariate matrices, <em>b</em> are the fixed-effects, <em>u</em> are the
random effects and <em>e</em> are the error terms. One way of defining the REML procedure is to:
<ul>
<li>Assign a flat prior to <em>b</em></li>.
<li>Integrate the likelihood with respect to both <em>b</em> and <em>u</em>.</li>
</ul>
The resulting REML likelihood is then maximized with respect to the variance parameters
(variances of <em>u</em> and <em>e</em>).<br>
<br>
To implement this in ADMB-RE we declare both <em>b</em> and <em>u</em> as random effects vectors.
As a result, the likelihood is integrated with respect to both parameters. But,
while there is a contribution to the objective function comming from the random effects distribution of <em>u</em>, 
there should be no such contribution from <em>b</em> (i.e. a flat prior).
<br><br>

Using this procedure you do not have to derive the "REML correction" to the log-likelihood  function by hand.
It is done automatically for you by ADMB-RE.


<h3><strong>Example: BCB Bowheads</strong></h3>
Punt and Zeh (2004, document SC/56/BRG1 presented to the SC of the IWC) combined data from two different counting
methodologies in order to estimate the rate of increase of BCB bowhead whales. On a log-scale they
fitted a linear mixed-effects models using REML estimates for the variance parameters. 
This example (<a href="bcb.tpl">bcb.tpl</a>) shows how <TT>random_effects</TT> can be used to obtain
REML estimates.
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